Computation of initial adjoint for NODE
Maybe later I can work through your math though I can say this.
First, a forward pass is done to obtain predictions of $z$, at every $t$. Then the adjoint state is run backward in time for every $t$. Which gives the learning impulse. So an initial run is done to obtain values of the dynamical system at all time points and the last of these is the initial point for the backward pass.
This is probably not enough to satisfy your question, hopefully, I can get more specific with respect to the math later.